弱非线性耦合二维各向异性谐振子的动力学行为

研究了弱非线性耦合二维各向异性谐振子的奇点稳定性及其在相空间中的轨迹.首先,求得弱非线性耦合二维各向异性谐振子的奇点;其次,分别利用Lyapunov间接法和梯度系统方法讨论该系统的平衡点稳定性;最后,用Matlab方法对系统进行数值模拟,并运用庞加赖截面观察系统在相空间的运动轨迹,发现随着能量的增加系统经历规则运动、规则运动与混沌并存等阶段,最后出现了混沌现象.     

基于控制系统直接方法的非线性系统预见控制器设计

针对一类非线性离散时间系统给出最优预见控制器设计方法. 首先运用非线性控制系统直接控制方法的思想, 将非线性反馈部分作为形式输入, 使得系统成为“形式上”的线性系统; 然后, 针对该线性系统, 利用最优预见控制的基本方法设计最优预见控制器; 最后, 利用形式输入与实际输入的关系得到非线性离散时间系统的最优预见控制器. 证明了如果形式线性系统满足一定的可镇定和可检测条件, 则闭环系统是渐近稳定的. 数值仿真结果表明了控制器的有效性.

     

非线性控制系统的全局输出调节

讨论了非线性控制系统的全局输出调节.首先推广精确线性化方法,通过状态反馈 和微分同胚将非线性系统的全局输出调节问题,转化为线性系统对非线性系统的跟踪问题. 通过提出可解性的概念,得到线性系统对非线性系统全局跟踪的条件,该结果是线性系统结 果的推广.在反馈同胚变换全局成立条件下,得到非线性控制系统全局输出调节问题的充分 条件,该条件对外部系统只做较弱的可解性假设,在反馈同胚变换局部成立的条件下,可得局 部结果.     

非线性控制系统的全局输出调节1)

讨论了非线性控制系统的全局输出调节.首先推广精确线性化方法,通过状态反馈和微分同胚将非线性系统的全局输出调节问题,转化为线性系统对非线性系统的跟踪问题.通过提出可解性的概念,得到线性系统对非线性系统全局跟踪的条件,该结果是线性系统结果的推广.在反馈同胚变换全局成立条件下,得到非线性控制系统全局输出调节问题的充分条件,该条件对外部系统只做较弱的可解性假设,在反馈同胚变换局部成立的条件下,可得局部结果.     

基于支持向量机α阶逆系统方法的非线性内模控制

为了提高传统逆系统方法的鲁棒性和抗干扰能力, 提出了基于支持向量机α阶逆系统方法的非线性内模控制新方法. 该方法利用支持向量机辨识非线性系统的α阶逆模型, 并将其串连在原系统之前得到复合的伪线性系统. 对求得的伪线性系统采用内模控制方法进行控制. 仿真结果证明了该方法的有效性. 理论分析和仿真结果均表明, 该方法不依赖于系统的模型, 且较一般的逆系统方法鲁棒稳定性好, 设计简单, 跟踪精度高, 是解决非线性系统控制的一种可行的理论方法.     

基于神经网络模型的最小预测误差非线性自适应控制器

主要研究基于神经网络模型的最小预测误差非线性自适应控制算法.利用神经网 络激励函数的分段局部线性近似,将基于神经网络的非线性系统一步前向预测控制转化为一 系列局部的线性预测控制问题.利用线性系统参数估计方法获得神经网络预测模型的参数估 计.在此基础上利用并联线性系统的预测控制方法设计全局收敛的非线性系统预测控制器.     

基于区间自组织映射的α阶逆系统控制研究*

为了提高α阶逆系统在控制中的鲁棒性,提出了区间自组织映射模型。根据定义的损失函数,利用梯度下降法得到新的模型竞争学习算法,并证明了该竞争算法的收敛性。利用区间自组织映射良好的逼近性能辨识非线性系统的α阶逆系统,并将其串联在原系统之前得到复合伪线性系统。仿真结果表明,该逆系统有较高的精度,逆控制器有良好的跟踪效果和较强的鲁棒性。     

不确定非线性系统的变结构鲁棒控制

本文利用几何方法研究了一类非线性系统的变结构控制问题,提出了具有不确定模态非 线性系统的滑动模态控制器设计方法,并用简单算例说明了该设计方法的可行性.     

基于驻留时间切换下离散切换系统的异步控制器设计

针对一类切换线性系统,本文提出了一种基于系统状态的驻留时间策略.这种切换策略不仅使异步状态反馈切换系统稳定,而且缩短了系统的运行时间.对于异步切换系统的稳定性和增益问题,本文主要的结论是在子系统运行期间Lyapunov函数允许增加,同时又没有的限制.通过利用基于系统状态的驻留时间策略,推导出了切换线性系统的控制器设计的充分条件.得出的结论也可以推广到非线性切换系统.本文中最后给出的算例用于说明该方法的有效性.     

并联混合有源滤波器逆系统解耦控制

针对并联混合有源滤波器(SHAPF)这一强耦合非线性系统的控制问题,提出了一种基于逆系统方法的SHAPF反馈线性化解耦控制策略.首先根据SHAPF非线性数学模型,采用逆系统方法生成其α阶积分逆系统,进而构造出解耦的伪线性系统,然后利用极点配置方法对伪线性系统进行综合,设计了系统的闭环控制器,并给出了系统零动态的镇定条件,保证了闭环控制系统的稳定性.最后仿真实验表明该控制策略能够有效消除电网中的谐波电流,并且与传统线性反馈-前馈控制策略相比,该控制策略具有更好的动静态性能.     

Existence of solutions for a singular system of nonlinear fractional differential equations

In this paper, we establish the existence of positive solutions for a singular system of nonlinear fractional differential equations. The differential operator is taken in the standard Riemann-Liouville sense. By using Green’s function and its corresponding properties, we transform the derivative systems into equivalent integral systems. The existence is based on a nonlinear alternative of Leray-Schauder type and Krasnoselskii’s fixed point theorem in a cone.     

On a method for studying stability of nonlinear dynamic systems

A method for studying the stability of the equilibrium points of linearized, nonlinear dynamic systems of arbitrary order is considered. The method is based on the fact that, due to the nature of the mutual arrangement of the trajectories of the corresponding linearized system, and the boundaries of some simply-connected, bounded neighborhood of its equilibrium point, one can judge the asymptotic stability and instability of both this point and the equilibrium point of the nonlinear system. Necessary and sufficient conditions of asymptotic stability and sufficient conditions of instability of equilibrium points of linear systems are given. Together with the theorems of the first Lyapunov method, these conditions determine the sufficient conditions of asymptotic stability and instability of equilibrium points of nonlinear systems. In some cases, the proposed conditions may turn out to be preferable to the known ones.     

Semi‐Global Robust Output Regulation for A Class of Singular Nonlinear Systems with Unknown Algebraic Equations

This paper considers semi‐global robust output regulation problem for a class of singular nonlinear systems whose algebraic equations are not precisely known. Since the algebraic equations are not known, the output regulation problem of singular nonlinear systems cannot be solved by directly reducing the singular nonlinear system into a normal nonlinear system. Based on internal model principle, we convert the robust output regulation problem of singular nonlinear systems into a robust stabilization problem of an augmented singular nonlinear system. The augmented singular nonlinear system is also with unknown algebraic equations. However, without transforming the singular nonlinear system into a normal nonlinear system, it is shown that the augmented singular nonlinear system can be semi‐globally stabilized by a high gain output feedback control law under some reasonable assumptions. Moreover, the semi‐global stabilization control law of the augmented singular nonlinear systems also solves the semi‐global robust output regulation problem of the original singular nonlinear system.     

Computing bifurcation points via characteristics gain loci

The authors show how to check the crossing on the imaginary axis by the eigenvalues of the linearized system of differential equations depending on a real parameter μ via feedback system theory. E. Hopf's theorem (1942) refers to a system of ordinary differential equations depending on the real parameter μ in which, when a single pair of complex conjugate eigenvalues of the linearized equations cross the imaginary axis under the parameter vibration, near this critical condition periodic orbits appear. The authors present simple formulas for both static (one eigenvalue zero) and dynamic or Hopf (a single pure imaginary pair) bifurcations, and show some singular conditions (degeneracies) by continuing the bifurcation curves in the steady-state manifold. The bifurcation curves and singular sets of an interesting chemical reactor which possesses multiplicity in the equilibrium solutions and in the Hopf bifurcation points are described     

参数曲线曲面实奇异点的计算

本文主要讨论了利用Grobner基理论对参数曲线（面）的奇异点进行判断和计算。如果曲线（面）存在奇异点,由定义可知它的导矢（法矢）等于0。因此,曲线（面）奇异点的判定就是方程组的求解问题。由Hilbert弱零点定理可知,若一组多项式方程无公共零点,则其生成理想约化的Grobner基为[1]。在计算时,首先根据Grobner基理论判断 曲线（面）是否存在奇异点。当存在奇异点时,利用区间算法对实奇异点进行隔离和迭代。在确定奇异点的存在性时,根据曲线（曲面）的导矢（法矢）方程的Grobner基直 接进行判断,而不需要求解非线性代数方程组。若曲线曲面存在奇异点,进一步采用区间方法对奇异点进行隔离以确定曲线段或曲面片的正则性。该方法可以得到参数曲线曲面的所有实奇异点且达到任意精度。     

Singular extremals on Lie groups

We investigate the space of singular curves associated to a distribution ofk-planes, or, what is the same thing, a nonlinear deterministic control system linear in controls. A singular curve is one for which the associated linearized system is not controllable. If a quadratic positive-definite cost function is introduced, then the corresponding optimal control problem is known as the sub-Riemannian geodesic problem. The original motivation for our work was the question “Are all sub-Riemannian minimizers smooth?” which is equivalent to the question “Are singular minimizers necessarily smooth?” Our main result concerns the singular curves for a class of homogeneous systems whose state spaces are compact Lie groups. We prove that for this class each singular curve lies in a lower-dimensional subgroup within which it is regular and we use this result to prove that all sub-Riemannian minimizers are smooth. A central ingredient of our proof is a symplectic-geometric characterization of singular curves formulated by Hsu. We extend this characterization to nonsmooth singular curves. We find that the symplectic point of view clarifies the situation and simplifies calculations.     

Tailored Finite Point Method for a Singular Perturbation Problem with Variable Coefficients in Two Dimensions

In this paper, we propose a tailored-finite-point method for a type of linear singular perturbation problem in two dimensions. Our finite point method has been tailored to some particular properties of the problem. Therefore, our new method can achieve very high accuracy with very coarse mesh even for very small ε, i.e. the boundary layers and interior layers do not need to be resolved numerically. In our numerical implementation, we study the classification of all the singular points for the corresponding degenerate first order linear dynamic system. We also study some cases with nonlinear coefficients. Our tailored finite point method is very efficient in both linear and nonlinear coefficients cases.     

Topological Numbers and Singularities in Scalar Images: Scale-Space Evolution Properties

Singular points of scalar images in any dimensions are classified by a topological number. This number takes integer values and can efficiently be computed as a surface integral on any closed hypersurface surrounding a given point. A nonzero value of the topological number indicates that in the corresponding point the gradient field vanishes, so the point is singular. The value of the topological number classifies the singularity and extends the notion of local minima and maxima in one-dimensional signals to the higher dimensional scalar images. Topological numbers are preserved along the drift of nondegenerate singular points induced by any smooth image deformation. When interactions such as annihilations, creations or scatter of singular points occurs upon a smooth image deformation, the total topological number remains the same.Our analysis based on an integral method and thus is a nonperturbative extension of the order-by-order approach using sets of differential invariants for studying singular points.Examples of typical singularities in one- and two-dimensional images are presented and their evolution induced by isotropic linear diffusion of the image is studied.     

Double SIB points in differential-algebraic systems

A singularity-induced bifurcation (SIB) describes the divergence of one eigenvalue through infinity when an equilibrium locus of a parameterized differential-algebraic equation (DAE) crosses a singular manifold. The present note extends the analysis of this behavior to cover double SIB points, for which two eigenvalues diverge. The key assumption supporting this phenomenon is that the Kronecker index jumps by two at the singularity. In this situation, double SIB points are shown to undergo generically a transition from a spiral to a saddle in the linearized problem, after restricting the analysis to the corresponding invariant subspace. Typical examples arise in the context of nonlinear RLC circuits. The setting for the study is that of semi-explicit DAEs in Hessenberg form with arbitrary index.